Alphabeta Math
CorollaryStatement: AI-adaptedProof: AI-adaptedprecheck passaudited 2026-07-31
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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

  • Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
  • AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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Every convergent real series is Cesaro summable and Abel summable to its ordinary sum

Statement

If ∑n≥0an converges ordinarily to s, then it is both Cesaro summable and Abel summable to s.

Facts & Assumptions

Given: Partial sums Sn→s.

[L1]

The Cesaro means of a convergent sequence converge to the same limit (If xk→L then σn→L: convergence implies (C,1)-summability to the same value).

[L2]

Abel's limit theorem sends an ordinarily convergent series to its ordinary sum (Abel's limit theorem: if a real series converges to s, then its power series tends to s as x↑1).

Proof

technique · direct
1.1

Apply [L1] to (Sn) to obtain σn→s, which is Cesaro summability.

givenL1
2.1

Apply [L2] to the original series to obtain Abel summability to s.

givenL2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources